Currently, I am reading John Hull's Options, Futures and Other Derivatives. On page 401, the author mentions the following:

Suppose that the delta of a call option on a stock is $0.6$, stock price is $\$100$ and the option price is $\$10$. Imagine an investor who has sold call options to buy $2,000$ shares of a stock. The investor’s position could be hedged by buying $$0.6 \times 2,000 = 1,200 \text{ shares}.$$ The gain (loss) on the stock position would then tend to offset the loss (gain) on the option position.

I do not understand the bold sentence.

My thought: As stock price increases while keeping other factors unchanged, the payoff for a call option increases, as the difference between terminal stock price and strike price increases. Thus the call option value increases. However, it is contradicting the bold sentence.

Furthermore, in this case, delta is a positive number $0.6$. Wouldn't this mean that an increase in underlying asset price leads to an increase in option value?


Let $C\left(S,t\right) $ represent the price of the call option when the underlying price is S at time t. Now if S changes by h instantaneously, the call price becomes $C\left(S+h, t\right) $. So the change in the call option price is:

$C\left(S+h, t\right) - C\left(S,t\right) $

Which you can approximate via first order Talyor series:

$C\left(S+h, t\right) - C\left(S,t\right) \approx \frac{\partial C}{\partial S}h$

The derivative on the right hand side is 0.6 in your question.

In summary, if the stock price changes by a small amount h, the price of the call option will change, resulting in a gain or loss (LHS), which will be offset by the position on the right hand side (RHS), which is 0.6 units of the stock per call option, or 1200 shares per 2000 call options .

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